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**What is Set Theory ?**

We define a set as an unordered collection of objects with no repeated elements and where the number of elements is finite. This is known as the Axiom of Extension.

Set theory is a branch of mathematics that deals with collections and the properties thereof. In other words, set theory is the study of mathematical figments called sets that describe collections of objects sharing certain properties.

We are about to embark on a journey that will take us from finite sets, to the rules for combining them, and finally to the beautiful analogies that describe the behavior of infinite sets. Although we may not use all the basic properties of these collections at once, we will be able to use many of them in combination. Set theory has been used in many fields of mathematics (particularly algebra, geometry and analysis) and is also used in computer science.

Sets are either finite or infinite. An infinite set has an indefinite number that cannot be counted because it goes on forever in both directions (as in the sets {1, 2} and {2, 3}) , while a finite set numbers are like {1, 2, 3},

The particular name Un that is used to denote the universal set is, in the English language, actually a misnomer. In the German language, this universal set was called überlegungsaula, which literally means “thinking room”. But since it became confused with Latin Universitas (university), it was decided to use the word Universe instead of Universal Set. This word Universe is derived from universum – “all things”, as opposed to singulum (single thing).

**Types Of Set Theory**

There are several different types of set theory and in in each , certain statements that can be made about sets are assumed to be axioms.

- Zermelo–Fraenkel set theory (ZF)- ZF is the most commonly used set theory. It was developed around 1908 by Ernst Zermelo and Abraham Fraenkel and subsequently refined by several mathematicians, including Thoralf Skolem, who added the “axiom of replacement” in 1922. A large portion of the mathematical community agrees to the fact that ZF is correct, thus making it a widely accepted theorem.
- Von Neumann–Bernays- NBG is a less widely used version of ZF. It was developed in the late 1920s by John von Neumann and his student Emmy Noether. Gödel showed that it was consistent with ZF, thus making NBG one of the most widely accepted theories of mathematics. Although elegant, NBG suffered from many problems and was not very useful in practice.
- Howard’s axiomatization of set theory (1918)- Howard’s axiomatization of set theory was proposed by a young mathematician named Alfred North Whitehead. It is the most commonly used axiomatization of set theory. The idea behind Howard’s axiomatization was to remove the concept of replacement from ZF and NBG by making them more like arithmetic than logic.
- Godel’s theory of the universe (1931-1936)- Alan Turing hypothesized that all mathematical objects can be translated into computable functions. It was proven by Kurt Gödel in 1931 and by Alonzo Church in 1936 that any system of logic capable of defining the truths of arithmetic is incomplete. This means many statements are true, but it cannot be proved. This insight led to a new form of mathematics that became known as post-Turing mathematics

**Importance Of Set Theory**

Set theory is helpful when mathematicians need to find information about fixed sets. It also helps to create mathematical items. For example, set theory helps to find elements that are not in a set, as well as find subsets of given sets.

The study of how sets behave is called “set structure”, and the branches of mathematics with interested in more theoretical questions are called “set theory”. It is concerned with set-theoretic axioms and laws; the results about sets can be organized into various categories, such as cardinality, power set or even beauty.

The treatment of the concept of the universal set by Zermelo and Fraenkel, who formalized ZF in 1908, helped to show that Euclidean geometry is independent of the axioms of set theory.

Set theory is also essential in understanding the concept of function, by describing the possible sets of values that a function can call for. This applies to numerous phenomena in mathematics, such as irrational numbers, which are not in the set of natural numbers.

Set theory is also applied to abstract algebra under the name “algebraic theory”.

set theory is also important in modern mathematics because it led to the concept of mathematical structure, which has led to the development of category theory.

**Topics Studies Under Set Theory As a Subject**

- Cardinality
- Power set
- Subsets

– The idea of a subset is defined by the following definition. For example, for the set {1,2,3}, the subset [1,2], is the same set as the set {1} and {2}.

- Union and intersection of sets;

-the union of sets and the intersection of sets are defined in the following mathematical definition.

- Cartesian product; Cross product
- Complete and incomplete intersections:
- Power set
- Topology
- Subset
– In this function one takes all subsets of a first set “S”, and increases their cardinality to form another random list.

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